11 HOMOGENEOUS SPACES 53 coset x = xH (resp. x = Hx) in G/H (resp. H\G) and each s eG, write s • i = C&x)H (resp. x - ^ = H(xy)). Then it is clear that G acts on the left (resp. on the right) on G/H (resp. H\G), and that this action is transitive. The set G/H (resp. H\G) together with this action of G is called the homogeneous space of left (resp. right) cosets ofH in G. We shall work throughout with G/H. Everything we shall do can be transposed in an obvious way to apply to H\G. Let n denote the canonical mapping x\-+ xH of G onto G/H. Clearly G/H is the set of orbits of elements of G for the right action (A, x) i-» xh (12.10.13) of H on G. If G is a topological group, we can therefore topologize G/H with the topology defined in (12.10), which is called the quotient by H of the topology of G. If x0 e G and if x0 = jc0H is its image in G/H, we obtain a fundamental system of neighborhoods of x0 in G/H by considering the canonical images in G/H of the neighborhoods V of x0 in G (i.e., for each V, the set of cosets xH of the elements x e V, or equivalently the image of V (or of VH) under the canonical mapping n : G -> G/H). Whenever we consider G/H as a topological space, it will always be this topology that is meant, unless the contrary is expressly stated. (12.11.1) The group G acts continuously on G/H. Let (s0 , x0) be a point of G x (G/H) and let x0 be an element of the coset x0. Every neighborhood of s0 • x0 is of the form ?r(V), where V is a neighborhood of ,y0 x0 in G. There exists a neighborhood U of s0 and a neighborhood W of x0 such that the relations s e U and x e W imply sx e V; hence the relations s e U and x e 7t(W) imply s - x e 7i(V). (12.11.2) Let G be a topological group, H a subgroup ofG. (i) G/H is Hausdorff if and only ifH is closed. (ii) G/H is discrete if and only ifH is open. (in) IfH is discrete, then every xeG has a neighborhood V such that the restriction of n to V is a homeomorphism of V onto the neighborhood n(V) of n(x) = x in G/H. (i) If G/H is Hausdorff, then (n(e)} is closed in G/H (12.3.4) and hence H = n~l(n(e)) is closed in G. Conversely, if H is closed in G, the set of pairs (x, y) e G x G belonging to the same orbit under the action of H on the right is the set of (x, y) such that x^lyeH, and is therefore closed, being the inverse image of H under the continuous mapping (x, y)t-+x~1y. Hence G/H is Hausdorff (12.10.8). (ii) If G/H is discrete, the set {n(e)} is open in G/H and therefore H = n~l(n(e)) is open in G. Conversely, if H is open in G, then so is eachd orbit ofx with respect to M is defined to be the set